I always assumed part of the reason was just to make life easier on the ballboys/ballgirls because they have to spend the entire game sitting so close to the crowd. Whereas the players are a little removed from the action, the ballboys would have to endure endless nights of "give us a ball, come on" and "you threw the last one over there" and "my kid isn't tall enough to catch them when you throw it up" etc.

It's a lot easier for the person to just say "It's policy that we don't give them out" then to have fans hounding them all night for balls that get hit down the line.